Process and device for displacing a moveable unit on a base

ABSTRACT

According to the invention:  
     a) a force (F) is determined which, applied to the moveable unit ( 4 ), produces a combined effect, on the one hand, on the moveable unit ( 4 ) so that it exactly carries out the envisaged displacement on the base ( 2 ), especially as regards the prescribed duration and prescribed distance of the displacement, and, on the other hand, on the elements (MA 1,  MA 2,  MA 3, 4 ) brought into motion by this displacement so that all these elements are immobile at the end of said displacement of the moveable unit ( 4 ); and  
     b) the force (F) thus determined is applied to the moveable unit ( 4 ).

BACKGROUND OF THE INVENTION

[0001] 1. Field of the Invention

[0002] The present invention relates to a process and a device fordisplacing a moveable unit on a base.

[0003] Said device is of the type comprising a controllable actuator,for example an electric motor, intended to give rise to a lineardisplacement of the moveable unit on the base, as well as a system whichis formed of a plurality of elements which are brought into motion uponthe displacement of said moveable unit.

[0004] Within the context of the present invention, said system exhibitsat least two different motions and comprises as elements which may bebrought into motion, in particular:

[0005] said base which can be mounted elastically with respect to thefloor, especially so as to isolate it from vibrations originating fromsaid floor; and/or

[0006] one or more auxiliary masses, for example measurement supportsand/or loads, which are tied elastically to the base; and/or

[0007] one or more auxiliary masses, for example likewise measurementsupports and/or loads, which are tied elastically to the moveable unit.

[0008] When the moveable unit is set into motion, said elements of thesystem begin to move. However, especially by reason of the aforesaidelastic link, these elements still continue to move when thedisplacement of the moveable unit has terminated and when the lattercomes to a stop.

[0009] Such a continuance of the motions of said system is generallyundesirable, since it may entail numerous drawbacks. In particular, itmay disturb measurements, especially positioning measurements, which aremade on the moveable unit or on these elements.

[0010] Also, an object of the present invention is to control themoveable unit in such a way that all the moving elements of said system,for example the base and/or auxiliary masses, are stationary at the endof the displacement of the moveable unit.

[0011] As regards said base, if it is mounted elastically with respectto the floor, it is known that, when the moveable unit is set intomotion, during the acceleration and deceleration phases, it is subjectedto the reaction of the force applied to the moveable unit by theactuator. This reaction load excites the base which then oscillates onits supports. This disturbs the relative positioning of the moveableunit with respect to the base, and greatly impedes the accuracy of thedevice.

[0012] This relative position error persists after the end of thedisplacement of the moveable unit and disappears only after thestabilization (which takes place much later) of the base.

[0013] Various solutions for remedying this drawback are known. Some ofthese solutions make provision in particular:

[0014] to immobilize the base during the acceleration and decelerationphases via a disabling system, for example an electromagnetic disablingsystem, which is mounted in parallel with the elastic supports. However,this known solution prevents the supports from isolating the base fromthe vibrations originating from the floor during said acceleration anddeceleration phases;

[0015] to cancel the effect produced by the force developed by theactuator, by making provision for an additional actuator which isarranged between the base and the floor and which develops an additionalforce of the same amplitude but oppositely directed; or

[0016] to displace an additional moveable unit on the base according toa similar displacement, but oppositely directed, with respect to thedisplacement of the moveable unit, so as to cancel the inertia effects.

[0017] However, none of these known solutions is satisfactory, sincetheir effectivenesses are restricted and since they all requiresupplementary means (disabling system, additional actuator, additionalmoveable unit) which increase in particular the complexity, the cost andthe bulkiness of the device.

[0018] Moreover, above all, these solutions implement an action whichacts only on the base and not on the other elements of the system which,for their part, continue to move when the moveable unit is stationary.

[0019] The object of the present invention is to remedy these drawbacks.It relates to a process for displacing, in an extremely accurate mannerand at restricted cost, a moveable unit on a base mounted for example onthe floor, whilst bringing all the motions to which this displacementgives rise to a stop at the end of the displacement, said moveable unitbeing displaced linearly according to a displacement which ispredetermined in terms of distance and time, under the action of acontrollable force.

[0020] Accordingly, said process is noteworthy according to theinvention in that:

[0021] a) equations are defined which:

[0022] illustrate a dynamic model of a system formed by elements, ofwhich said moveable unit is one, which are brought into motion upon adisplacement of said moveable unit; and

[0023] comprise at least two variables, of which the position of saidmoveable unit is one;

[0024] b) all the variables of this system, together with said force,are expressed as a function of one and the same intermediate variable yand of a specified number of derivatives as a function of time of thisintermediate variable, said force being such that, applied to saidmoveable unit, it displaces the latter according to said specifieddisplacement and renders all the elements of said system immobile at theend of said displacement;

[0025] c) the initial and final conditions of all said variables aredetermined;

[0026] d) the value as a function of time of said intermediate variableis determined from the expressions for the variables defined in step b)and said initial and final conditions;

[0027] e) the value as a function of time of said force is calculatedfrom the expression for the force, defined in step b) and said value ofthe intermediate variable, determined in step d); and

[0028] f) the value thus calculated of said force is applied to saidmoveable unit.

[0029] Thus, the force applied to the moveable unit enables the latterto carry out the predetermined displacement envisaged, especially interms of time and distance, whilst rendering the elements brought intomotion by this displacement immobile at the end of the displacement sothat they do not oscillate and, in particular, do not disturb therelative positioning between themselves and the moveable unit.

[0030] It will be noted moreover that, by reason of this combinedcontrol of said moveable unit and of said moving elements, one obtainsan extremely accurate displacement of the moveable unit in a referenceframe independent of the base and tied for example to the floor.

[0031] It will be noted that the implementation of the process inaccordance with the invention is not limited to a displacement along asingle axis, but can also be applied to displacements along several axeswhich can be regarded as independent.

[0032] Advantageously, in step a), the following operations are carriedout: the variables of the system are denoted xi, i going from 1 to p, pbeing an integer greater than or equal to 2, and the balance of theforces and of the moments is expressed, approximating to first order ifnecessary, in the so-called polynomial matrix form:

A(s)X=bF

[0033] with:

[0034] A(s) matrix of size p×p whose elements Aij(s) are polynomials ofthe variable s=d/dt;

[0035] X the vector $\begin{pmatrix}{x1} \\\vdots \\{xp}\end{pmatrix};$

[0036] b the vector of dimension p; and

[0037] F the force exerted by the motor.

[0038] Advantageously, in step b), the following operations are carriedout:

[0039] the different variables xi of said system, i going from 1 to p,each being required to satisfy a first expression of the form:${{xi} = {\sum\limits_{j = 0}^{j = r}{pi}}},{j \cdot y^{(j)}},$

[0040] the y^((j)) being the derivatives of order j of the intermediatevariable y, r being a predetermined integer and the pi, j beingparameters to be determined, a second expression is obtained by puttingy^((j))=s^(j).y:${{xi} = {{\left( {{\sum\limits_{j = 0}^{j = r}{pi}},{j \cdot s^{j}}} \right)y} = {{{Pi}(s)} \cdot y}}},$

[0041] a third expression of vectorial type is defined on the basis ofthe second expressions relating to the different variables xi of thesystem:

[0042] comprising the vector X = P ⋅ y $P = \begin{pmatrix}{P1} \\\vdots \\{Pp}\end{pmatrix}$

[0043] said vector P is calculated, by replacing X by the value P.y inthe following system: $\left\{ \begin{matrix}{{B^{T} \cdot {A(s)} \cdot {P(s)}} = {{Op} - 1}} \\{{{{bp} \cdot F} = {\sum\limits_{j = i}^{j = p}{Ap}}},\quad {{j(s)} \cdot {{Pj}(s)} \cdot y}}\end{matrix} \right.$

[0044] in which:

[0045] B^(T) is the transpose of a matrix B of size px(p−1), such thatB^(T)b=Op−1;

[0046] bp is the p-th component of the vector b previously defined; and

[0047] Op−1 is a zero vector of dimension (p−1);

[0048] the values of the different parameters pi,j are deduced from thevalue thus calculated of the vector P; and

[0049] from these latter values are deduced the values of the variablesxi as a function of the intermediate variable y and of its derivatives,on each occasion using the corresponding first expression.

[0050] Thus, a fast and general method of calculation is obtained forcalculating the relations between the variables of the system and saidintermediate variable, in the form of linear combinations of the latterand of its derivatives with respect to time.

[0051] Advantageously, in step d), a polynomial expression for theintermediate variable y is used to determine the value of the latter.

[0052] In this case, preferably, the initial and final conditions of thedifferent variables of the system, together with the expressions definedin step b), are used to determine the parameters of this polynomialexpression.

[0053] In a first embodiment, for displacing a moveable unit on a basewhich is mounted elastically with respect to the floor and which may besubjected to linear and angular motions, advantageously, the variablesof the system are the linear position x of the moveable unit, the linearposition xB of the base and the angular position θz of the base, whichsatisfy the relations: $\left\{ \begin{matrix}{x\quad = {y + {\left( {\frac{r\quad B}{k\quad B} + \frac{r\quad \theta}{k\quad \theta}} \right)y^{(1)}} + {\left( {\frac{m\quad B}{k\quad B} + \frac{r\quad B\quad r\quad \theta}{k\quad B\quad k\quad \theta} + \frac{J}{k\quad \theta}} \right)y^{(2)}} +}} \\{{\left( {\frac{r\quad B\quad J}{k\quad B\quad k\quad \theta} + \frac{m\quad B\quad r\quad \theta}{k\quad B\quad k\quad \theta}} \right)y^{(3)}} + {\frac{m\quad B\quad J}{k\quad B\quad k\quad \theta}y^{(4)}}} \\{{x\quad B}\quad = {{- \frac{m}{k\quad B}}\left( {{\frac{J}{k\quad \theta}y^{(4)}} + {\frac{r\quad \theta}{k\quad \theta}y^{(3)}} + y^{(2)}} \right)}} \\{{\theta \quad z}\quad = {{- d}\frac{m}{k\quad \theta}\left( {{\frac{m\quad B}{k\quad B}y^{(4)}} + {\frac{r\quad B}{k\quad B}y^{(3)}} + y^{(2)}} \right)}}\end{matrix} \right.$

[0054] in which:

[0055] m is the mass of the moveable unit;

[0056] mB, kB, kθ, rB, rθ are respectively the mass, the linearstiffness, the torsional stiffness, the linear damping and the torsionaldamping of the base;

[0057] J is the inertia of the base with respect to a vertical axis;

[0058] d is the distance between the axis of translation of the centerof mass of the moveable unit and that of the base; and

[0059] y⁽¹⁾, y⁽²⁾, y⁽³⁾ and y⁽⁴⁾ are respectively the first to fourthderivatives of the variable y.

[0060] This first embodiment makes it possible to remedy the aforesaiddrawbacks (inaccurate displacement, etc) related to the setting of thebase into oscillation during the displacement of the moveable unit.

[0061] In a second embodiment, for displacing on a base a moveable uniton which are elastically mounted a number p of auxiliary masses MAi, pbeing greater than or equal to 1, i going from 1 to p, advantageously,the variables of the system are the position x of the moveable unit andthe (linear) positions zi of the p auxiliary masses MAi, which satisfythe relations: $\left\{ \begin{matrix}{x\quad = {\left( {\prod\limits_{i = 1}^{p}\left( {{\frac{m\quad i}{ki}s^{2}} + {\frac{ri}{ki}s} + 1} \right)} \right) \cdot y}} \\{{zi}\quad = {\left( {\prod\limits_{\substack{j = 1 \\ j \neq \quad i}}^{p}\left( {{\frac{m\quad i}{kj}s^{2}} + {\frac{ri}{kj}s} + 1} \right)} \right) \cdot \left( {{\frac{r\quad i}{k\quad i}s} + 1} \right) \cdot y}}\end{matrix} \right.$

[0062] in which:

[0063] Π illustrates the product of the associated expressions;

[0064] mi, zi, ki and ri are respectively the mass, the position, thestiffness and the damping of an auxiliary mass MAi;

[0065] mj, kj and rj are respectively the mass, the stiffness and thedamping of an auxiliary mass MAj; and

[0066] s=d/dt.

[0067] In a third embodiment, for displacing a moveable unit on a basewhich is mounted elastically with respect to the floor and on which iselastically mounted an auxiliary mass, advantageously, the variables ofthe system are the positions x, xB and zA respectively of the moveableunit, of the base and of the auxiliary mass, which satisfy therelations: $\left\{ \begin{matrix}{x\quad = {\left\lbrack {{\left( {{m\quad A\quad s^{2}} + {r\quad A\quad s} + {k\quad A}} \right) \cdot \left( {{m\quad B\quad s^{2}} + {\left( {{r\quad A} + {r\quad B}} \right)s} + \left( {{k\quad A} + {k\quad B}} \right)} \right)} - \left( {{r\quad A\quad s} + {k\quad A}} \right)^{2}} \right\rbrack \cdot y}} \\{{x\quad B}\quad = {{- M}\quad y^{(2)}}} \\{{z\quad A}\quad = {- {M\left( {{r\quad A\quad y^{(3)}} + {k\quad A\quad y^{(2)}}} \right)}}}\end{matrix} \right.$

[0068] in which:

[0069] M, mB and mA are the masses respectively of the moveable unit, ofthe base and of the auxiliary mass;

[0070] rA and rB are the dampings respectively of the auxiliary mass andof the base;

[0071] kA and kB are the stiffnesses respectively of the auxiliary massand of the base; and

[0072] s=d/dt.

[0073] In a fourth embodiment, for displacing on a base mountedelastically with respect to the floor, a moveable unit on which iselastically mounted an auxiliary mass, advantageously, the variables ofthe system are the positions x, xB and zC respectively of the moveableunit, of the base and of the auxiliary mass, which satisfy therelations: $\left\{ \begin{matrix}{x\quad = {\left\lbrack {\left( {{m\quad C\quad s^{2}} + {r\quad C\quad s} + {k\quad C}} \right) \cdot \left( {{m\quad B\quad s^{2}} + {r\quad B\quad s} + {k\quad B}} \right)} \right\rbrack \cdot y}} \\{{x\quad B}\quad = {\left\lbrack {{\left( {{m\quad C\quad s^{2}} + {r\quad C\quad s} + {k\quad C}} \right) \cdot \left( {{M\quad s^{2}} + {r\quad C\quad s} + {k\quad C}} \right)} - \left( {{r\quad C\quad s} + {k\quad C}} \right)^{2}} \right\rbrack \cdot y}} \\{{z\quad C}\quad = {\left( {{r\quad C\quad s} + {k\quad C}} \right) \cdot \left( {{m\quad B\quad s^{2}} + {r\quad B\quad s} + {k\quad B}} \right) \cdot y}}\end{matrix} \right.$

[0074] in which:

[0075] M, mB and mC are the masses respectively of the moveable unit, ofthe base and of the auxiliary mass;

[0076] rB and rC are the dampings respectively of the base and of theauxiliary mass;

[0077] kB and kC are the stiffnesses respectively of the base and of theauxiliary mass; and

[0078] s=d/dt.

[0079] The present invention also relates to a device of the typecomprising:

[0080] a base mounted directly or indirectly on the floor;

[0081] a moveable unit which may be displaced linearly on said base; and

[0082] a controllable actuator able to apply a force to said moveableunit with a view to its displacement on said base.

[0083] According to the invention, said device is noteworthy in that itfurthermore comprises means, for example a calculator:

[0084] which implement steps a) to e) of the aforesaid process, so as tocalculate a force which, applied to said moveable unit, makes itpossible to obtain the combined effect or control indicated above; and

[0085] which determine a control command and transmit it to saidactuator so that it applies the force thus calculated to said moveableunit, during a displacement.

[0086] Thus, over and above the aforesaid advantages, the device inaccordance with the invention does not require any additional mechanicalmeans, thereby reducing its cost and its bulkiness and simplifying itsembodiment, with respect to the known and aforesaid devices.

[0087] The figures of the appended drawing will elucidate the manner inwhich the invention may be embodied. In these figures, identicalreferences designate similar elements.

[0088]FIGS. 1 and 2 respectively illustrate two different embodiments ofthe device in accordance with the invention.

[0089] FIGS. 3 to 7 represent graphs which illustrate the variationsover time of variables of the system, for a first embodiment of thedevice in accordance with the invention.

[0090] FIGS. 8 to 13 represent graphs which illustrate the variationsover time of variables of the system, for a second embodiment of thedevice in accordance with the invention.

[0091] The device 1 in accordance with the invention and representeddiagrammatically in FIGS. 1 and 2, according to two differentembodiments, is intended for displacing a moveable unit 4, for example amoveable carriage, on a base 2, in particular a test bench.

[0092] This device 1 can for example be applied to fast XY tables usedin microelectronics, to machine tools, to conveyors, to robots, etc.

[0093] In a known manner, said device 1 comprises, in addition to thebase 2 and to the moveable unit 4:

[0094] supports 3, of known type, arranged between the base 2 and thefloor S;

[0095] means (not represented), for example a rail, fixed on the base 2and enabling the moveable unit 4 to be displaced linearly on said base2; and

[0096] a controllable actuator 5, preferably an electric motor, able toapply a force F to said moveable unit 4 with a view to its displacementon the base 2.

[0097] Within the context of the present invention, the device 1comprises a system S1, S2 which is formed of various elements specifiedhereinbelow and variables according to the embodiment contemplated,which are brought into motion upon the displacement of the moveable unit4.

[0098] According to the invention, said device 1 is improved in such away as to obtain directly at the end of a displacement of the moveableunit 4:

[0099] accurate positioning of the latter in a reference frame (notrepresented), independent of the moveable unit 4 and of the base 2 andtied for example to the floor; and

[0100] immobilization of all the moving elements of said system S1, S2.

[0101] To do this, the device 1 moreover comprises, according to theinvention, calculation means 6 which calculate a particular force F,which is intended to be transmitted in the form of a control command tothe actuator 5, as illustrated by a link 7, and which is such that,applied to said moveable unit 4, it produces a combined effect (andhence combined control):

[0102] on the one hand, on the moveable unit 4 so that it exactlycarries out the envisaged displacement, especially as regards theprescribed duration and prescribed distance of displacement; and

[0103] on the other hand, on said system S1, S2 so that all its movingelements are immobile at the end of the displacement of the moveableunit 4.

[0104] Accordingly, said calculation means 6 implement the process inaccordance with the invention, according to which:

[0105] a) equations are defined which:

[0106] illustrate a dynamic model of said system (for example S1 or S2)formed by the different elements, of which said moveable unit 4 is one,which are brought into motion upon a displacement of said moveable unit4; and

[0107] comprise at least three variables, of which the position of saidmoveable unit 4 is one;

[0108] b) all the variables of this system, together with said force F,are expressed as a function of one and the same intermediate variable yand of a specified number of derivatives as a function of time of thisintermediate variable, said force F being required to be such that,applied to said moveable unit 4, it displaces the latter according tosaid specified displacement and renders all the elements of said systemimmobile at the end of said displacement;

[0109] c) the initial and final conditions of all said variables aredetermined;

[0110] d) the value as a function of time of said intermediate variableis determined from the expressions for the variables defined in step b)and said initial and final conditions; and

[0111] e) the value of said force is calculated from the expression forthe force, defined in step b) and said value of the intermediatevariable, determined in step d).

[0112] Thus, by virtue of the invention, the force F applied to themoveable unit 4 enables the latter to carry out the predetermineddisplacement envisaged, especially in terms of time and distance, whilstrendering the elements (specified hereinbelow) which are brought intomotion by this displacement immobile at the end of the displacement sothat they do not oscillate and, in particular, do not disturb therelative positioning between themselves and the moveable unit 4.

[0113] It will be noted moreover that, by reason of this combined effector control of said moveable unit 4 and of said moving elements, oneobtains an extremely accurate displacement of the moveable unit 4 in areference frame independent of the base 2 and tied for example to thefloor S.

[0114] Of course, the implementation of the present invention is notlimited to a displacement along a single axis, but can also be appliedto displacements along several axes which can be regarded asindependent.

[0115] According to the invention, in step d), a polynomial expressionfor the intermediate variable y is used to determine the value of thelatter, and the initial and final conditions of the different variablesof the system, together with the expressions defined in step b) are usedto determine the parameters of this polynomial expression.

[0116] The process in accordance with the invention will now bedescribed in respect of four different systems (of moving elements).

[0117] In a first embodiment (not represented), the supports 3 are ofelastic type and make it possible to isolate the base 2 from thevibrations originating from said floor S. The natural frequency of thebase 2 on said elastic supports 3 is generally a few Hertz. Furthermore,in addition to the translational motion of the moveable unit 4controlled by the force F, an angular motion is created between the base2 and the moveable unit 4. Specifically, in this case, the axis of themoveable unit 4 does not pass through its center of mass, the forceproduced by the actuator 5 creates a moment about the vertical axis. Therail is assumed to be slightly flexible and thus allows the moveableunit 4 small rotational motions about the vertical axis, whichcorresponds to the aforesaid relative angular motion between the base 2and the moveable unit 4.

[0118] Consequently, in this first embodiment, to displace the moveableunit 4 on the base 2 which is mounted elastically with respect to thefloor and which may be subjected to a (relative) angular motion, thevariables of the system are the linear position x of the moveable unit4, the linear position xB of the base 2 and the angular position θz ofthe base 2, which satisfy the relations: $\left\{ \begin{matrix}{x\quad = {y + {\left( {\frac{r\quad B}{k\quad B} + \frac{r\quad \theta}{k\quad \theta}} \right)y^{(1)}} + {\left( {\frac{m\quad B}{k\quad B} + \frac{r\quad B\quad r\quad \theta}{k\quad B\quad k\quad \theta} + \frac{J}{k\quad \theta}} \right)y^{(2)}} +}} \\{{\left( {\frac{r\quad B\quad J}{k\quad B\quad k\quad \theta} + \frac{m\quad B\quad r\quad \theta}{k\quad B\quad k\quad \theta}} \right)y^{(3)}} + {\frac{m\quad B\quad J}{k\quad B\quad k\quad \theta}y^{(4)}}} \\{{x\quad B}\quad = {{- \frac{m}{k\quad B}}\left( {{\frac{J}{k\quad \theta}y^{(4)}} + {\frac{r\quad \theta}{k\quad \theta}y^{(3)}} + y^{(2)}} \right)}} \\{{\theta \quad z}\quad = {{- d}\frac{m}{k\quad \theta}\left( {{\frac{m\quad B}{k\quad B}y^{(4)}} + {\frac{r\quad B}{k\quad B}y^{(3)}} + y^{(2)}} \right)}}\end{matrix} \right.$

[0119] in which

[0120] m is the mass of the moveable unit 4;

[0121] mB, kB, kθ, rB, rθ are respectively the mass, the linearstiffness, the torsional stiffness, the linear damping and the torsionaldamping of the base 2;

[0122] J is the inertia of the base 2 with respect to a vertical axis;

[0123] d is the distance between the axis of translation of the centerof mass of the moveable unit 4 and that of the base 2; and

[0124] y⁽¹⁾, y⁽²⁾, y⁽³⁾ and y⁽⁴⁾ are respectively the first to fourthderivatives of the variable y.

[0125] Specifically, in this first embodiment, the balance of the forcesand of the moments, the angle θz being approximated to first order, maybe written: $\begin{matrix}\left\{ \begin{matrix}{\quad {{mx}^{(2)} = F}} \\{\quad {{mBxB}^{(2)} = {{- F} - {kBxB} - {rBxB}^{(1)}}}} \\{\quad {{J\quad \theta \quad z^{(2)}} = {{- {dF}} - {k\quad {\theta\theta}\quad z} - {r\quad {\theta\theta}\quad z^{(1)}}}}}\end{matrix} \right. & (1)\end{matrix}$

[0126] It will be noted that, within the context of the presentinvention, α^((β)) is the derivative of order β with respect to time ofthe parameter α, regardless of α. Thus, for example, x⁽¹⁾ is the firstderivative of x with respect to time.

[0127] The calculation of the intermediate variable y is achieved byputting ${s = \frac{}{t}},$

[0128] x=P(s)y, xB=PB(s)y, θz=Pθ(s)y and by rewriting the system (1)with this notation: $\left\{ \begin{matrix}{\quad {{{ms}^{2}{P(s)}y} = F}} \\{\quad {{\left( {{mBs}^{2} + {rBs} + {kB}} \right){{PB}(s)}y} = {- F}}} \\{\quad {{\left( {{Js}^{2} + {r\quad \theta \quad s} + {k\quad \theta}} \right)P\quad {\theta (s)}y} = {- {dF}}}}\end{matrix}\quad \right.$

[0129] i.e.:${\left( {{mBs}^{2} + {rBs} + {kB}} \right){{PB}(s)}} = {{\frac{1}{d}\left( {{Js}^{2} + {r\quad \theta \quad s} + {k\quad \theta}} \right)P\quad \theta \quad (s)} = {{- {ms}^{2}}{P(s)}}}$

[0130] and hence: $\left\{ {\begin{matrix}{\quad {{P(s)} = {\left( {{\frac{mB}{kB}s^{2}} + {\frac{rB}{kB}s} + 1} \right)\left( {{\frac{J}{k\quad \theta}s^{2}} + {\frac{r\quad \theta}{k\quad \theta}s} + 1} \right)}}} \\{\quad {{{PB}(s)} = {{- \frac{m}{kB}}{s^{2}\left( {{\frac{J}{k\quad \theta}s^{2}} + {\frac{r\quad \theta}{k\quad \theta}s} + 1} \right)}}}} \\{\quad {{P\quad \theta \quad (s)} = {{- d}\frac{m}{k\quad \theta}{s^{2}\left( {{\frac{mB}{kB}s^{2}} + {\frac{rB}{kB}s} + 1} \right)}}}}\end{matrix}\quad} \right.$

[0131] From these expressions, we immediately deduce: $\begin{matrix}{{x = {\left( {{\frac{mB}{kB}s^{2}} + {\frac{rB}{kB}s} + 1} \right)\left( {{\frac{J}{k\quad \theta}s^{2}} + {\frac{r\quad \theta}{k\quad \theta}s} + 1} \right)y}}\left\{ \begin{matrix}{\quad {x = {y + {\left( {\frac{rB}{kB} + \frac{r\quad \theta}{k\quad \theta}} \right)y^{(1)}} + {\left( {\frac{mB}{kB} + \frac{{rBr}\quad \theta}{{kBk}\quad \theta} + \frac{J}{k\quad \theta}} \right)y^{(2)}} + {\left( {\frac{rBJ}{{kBk}\quad \theta} + \frac{{mBr}\quad \theta}{{kBk}\quad \theta}} \right)y^{(3)}} + {\frac{mBJ}{{kBk}\quad \theta}y^{(4)}}}}} \\{\quad {{xB} = {{- \frac{m}{kB}}\left( {{\frac{J}{k\quad \theta}y^{(4)}} + {\frac{r\quad \theta}{k\quad \theta}y^{(3)}} + y^{(2)}} \right)}}} \\{\quad {{\theta \quad z} = {{- d}\frac{m}{k\quad \theta}\left( {{\frac{mB}{kB}y^{(4)}} + {\frac{rB}{kB}y^{(3)}} + y^{(2)}} \right)}}}\end{matrix} \right.} & (2)\end{matrix}$

[0132] The expression for y as a function of x, x⁽¹⁾, xB, xB⁽¹⁾, θz andθz⁽¹⁾ is obtained by inversion. However, this formula is not necessaryin order to plan the trajectories of x, xB and θz. Specifically, sincewe want a stop-stop displacement of the moveable unit 4 between x0 atthe instant t0 and x1 at the instant t1, with

x ⁽¹⁾(t 0)=0=x ⁽¹⁾(t 1) and xB(t 0)=0=xB(t 1), xB ⁽¹⁾(t 0)=0=xB ⁽¹⁾(t 1)and θz(t 0)=0=θz(t 1), θz ⁽¹⁾(t 0)=0=θz ⁽¹⁾(t 1), with in addition F(t0)=0=F(t 1),

[0133] we deduce therefrom through the aforesaid expressions (2) thaty(t0)=x0,y(t1)=x1 and

y ⁽¹⁾(ti)=y ⁽²⁾(ti)=y ⁽³⁾(ti)=y ⁽⁴⁾(ti)=y ⁽⁵⁾(ti)=y ⁽⁶⁾(ti)=0, i=0.1

[0134] i.e. 14 initial and final conditions.

[0135] It is sufficient to choose y as a polynomial with respect to timeof the form: $\begin{matrix}{{y(t)} = {{x0} + {\left( {{x1} - {x0}} \right)\left( {\sigma (t)} \right)^{\alpha}{\sum\limits_{i = 0}^{\beta}{{ai}\left( {\sigma (t)} \right)}^{i}}}}} & (3)\end{matrix}$

[0136] with ${\sigma (t)} = \frac{t - {t0}}{{t1} - {t0}}$

[0137] and α≧7 and β≧6. The coefficients a0, . . . , aβ are thenobtained, according to standard methods, by solving a linear system.

[0138] The reference trajectory sought for the displacement of themoveable unit 4 is then given by expressions (2) with y(t) given byexpression (3).

[0139] Moreover, the force F as a function of time to be applied to themeans 5 is obtained by integrating the value of y obtained viaexpression (3) in the expression F(t)=M.x⁽²⁾(t).

[0140] In this first embodiment, we obtain: $\begin{matrix}{{F(t)} = {M\left\lbrack {y^{2} + {\left( {\frac{rB}{kB} + \frac{r\quad \theta}{k\quad \theta}} \right)y^{(3)}} + {\left( {\frac{mB}{kB} + \frac{{rBr}\quad \theta}{{kBk}\quad \theta} + \frac{J}{k\quad \theta}} \right)y^{(4)}} + {\left( \frac{{rBJ} + {{mBr}\quad \theta}}{{kBk}\quad \theta} \right)y^{(5)}} + {\frac{mBJ}{{kBk}\quad \theta}y^{(6)}}} \right\rbrack}} & \text{(3A)}\end{matrix}$

[0141] with y(t) given by expression (3).

[0142] Thus, since by virtue of the device 1 the base 2 is immobilizedat the end of the displacement, it does not disturb the positioning ofthe moveable unit 4 in the aforesaid reference frame so that saidmoveable unit 4 is positioned in a stable manner as soon as itsdisplacement ends. Moreover, since its displacement is carried out in anaccurate manner, its positioning corresponds exactly in said referenceframe to the sought-after positioning.

[0143] Represented in FIGS. 3 to 7 are the values respectively of saidvariables y (in meters m), x (in meters m), xB (in meters m), θz (inradians rd) and F (in Newtons N) as a function of time t (in seconds s)for a particular exemplary embodiment, for which:

[0144] m=40 kg;

[0145] mB=800 kg;

[0146] kB=mB(5.2π)² corresponding to a natural frequency of 5 Hz;

[0147] rB=0.3{square root}{square root over (kBmB)} corresponding to anormalized damping of 0.3;

[0148] J=120 Nm corresponding to the inertia of the moveable unit 4;

[0149] kθ=J(10.2π)² corresponding to a natural rotational frequency of10 Hz;

[0150] r·=0.03{square root}{square root over (kθJ)} corresponding to anormalized rotational damping of 0.3;

[0151] d=0.01 m corresponding to the off-centering of the moveable unit4;

[0152] t1−t0=0.4 s; and

[0153] x1−x0=25 mm.

[0154] The moveable unit 4 is displaced from the position x0 at rest (x0⁽¹⁾=0) at the instant t0, to the position x1 at rest (x1 ⁽¹⁾=0) at theinstant t1. It is therefore displaced over a distance of 25 mm in 0.4 s.To obtain this displacement, as well as the immobilization (at the endof said displacement) of the various motions to which the displacementgives rise, the force F represented in FIG. 7 should be applied to saidmoveable unit 4. This force is given by expression (3A) with y given by(3) for α=7 and β=6. In this case, the coefficients a0 up to a6 aregiven by a0=1716, a1=−9009, a2=20020, a3=−24024, a4=16380, a5=−6006,a6=924.

[0155] In a second embodiment represented in FIG. 1, the system S1comprises, in addition to the moveable unit 4, a number p of auxiliarymasses MAi, p being greater than or equal to 1, i going from 1 to p,which are linked respectively by elastic links e1 to ep of standardtype, in particular springs, to said moveable unit 4. In the examplerepresented, p=3.

[0156] In this case, the variables of the system are the position x ofthe moveable unit 4 and the positions zi of the p auxiliary masses MAi,which satisfy the relations: $\begin{matrix}\left\{ \begin{matrix}{\quad {x = {\left( {\prod\limits_{i = 1}^{p}\left( {{\frac{mi}{ki}s^{2}} + {\frac{ri}{ki}s} + 1} \right)} \right) \cdot y}}} \\{\quad {{zi} = {\left( {\underset{j \neq i}{\overset{p}{\prod\limits_{j = 1}}}\left( {{\frac{mj}{kj}s^{2}} + {\frac{rj}{kj}s} + 1} \right)} \right) \cdot \left( {{\frac{ri}{ki}s} + 1} \right) \cdot y}}}\end{matrix} \right. & (4)\end{matrix}$

[0157] in which:

[0158] Π illustrates the product of the associated expressions;

[0159] mi, zi, ki and ri are respectively the mass, the position, thestiffness and the damping of an auxiliary mass MAi;

[0160] mj, kj and rj are respectively the mass, the stiffness and thedamping of an auxiliary mass MAj; and −s=d/dt.

[0161] Specifically, the dynamic model of the system S1 may be written:$\begin{matrix}\left\{ \begin{matrix}{{Mx}^{(2)} = {F + {\sum\limits_{i = 1}^{p}\quad \left( {{{ki}\left( {{zi} - x} \right)} + {{ri}\left( {{zi}^{(1)} - x^{(1)}} \right)}} \right)}}} \\{{{Mizi}^{(2)} = {{{ki}\left( {x - {zi}} \right)} + {{ri}\left( {x^{(1)} - {zi}^{(1)}} \right)}}},{i = 1},{{\ldots p}.}}\end{matrix} \right. & (5)\end{matrix}$

[0162] As in the foregoing, we wish to find laws of motion which ensurethe desired displacement of the moveable unit 4, the auxiliary massesMAi (for example measurement devices and/or loads) being immobilized assoon as the moveable unit 4 stops.

[0163] Accordingly, the intermediate variable y is calculated by thesame approach as earlier and the trajectory of the moveable unit 4 isplanned by way thereof.

[0164] The intermediate variable y being required to satisfy x=P(s)y,zi=Pi(s)y, i=1, . . . , p, with ${s = \frac{}{t}},$

[0165] we must have, substituting these relations into the system (5):

(mis ² +ris+ki)Pi=(ris+ki)P, i=1, . . . , p

[0166] From this expression, we immediately derive:${{P(s)} = \left( {\prod\limits_{i = 1}^{p}\quad \left( {{\frac{mi}{ki}s^{2}} + {\frac{ri}{ki}s} + 1} \right)} \right)},{{Pi} = {\left( {\prod\limits_{\underset{{j \neq i}\quad}{j = 1}}^{p}\quad \left( {{\frac{mj}{kj}s^{2}} + {\frac{rj}{kj}s} + 1} \right)} \right)\left( {{\frac{ri}{ki}s} + 1} \right)}},$

[0167] thereby proving the aforesaid formulae (4).

[0168] In this case, it may be demonstrated that the force F to beapplied satisfies the relation:${F(t)} = {\left\lbrack {{\left( {{Ms}^{2} + {\left( {\sum\limits_{j = 1}^{p}\quad {rj}} \right)s} + \left( {\sum\limits_{j = 1}^{p}\quad {kj}} \right)} \right){\prod\limits_{i = 1}^{p}\quad \left( {{\frac{mi}{ki}s^{2}} + {\frac{ri}{ki}s} + 1} \right)}} - {\sum\limits_{i = 1}^{p}\quad {\left( {{ris} + {ki}} \right){\prod\limits_{\underset{{j \neq i}\quad}{j = 1}}^{p}\quad \left( {{\frac{mj}{kj}s^{2}} + {\frac{rj}{kj}s} + 1} \right)}}}} \right\rbrack {y.}}$

[0169] The aforesaid formulae are verified and specified hereinbelow fortwo and three auxiliary masses MAi respectively.

[0170] In the case of two auxiliary masses (p=2), the model may bewritten: $\quad\left\{ \begin{matrix}{{Mx}^{(2)} = {F - {{k1}\left( {x - {z1}} \right)} - {{r1}\left( {x^{(1)} - {z1}^{(1)}} \right)} - {{k2}\left( {x - {z2}} \right)} - {{r2}\left( {x^{(1)} - {z2}^{(1)}} \right)}}} \\{\quad {{m1z1}^{(2)} = {{{k1}\left( {x - {z1}} \right)} + {{r1}\left( {x^{(1)} - {z1}^{(1)}} \right)}}}} \\{\quad {{m2z2}^{(2)} = {{{k2}\left( {x - {z2}} \right)} + {{r2}\left( {x^{(1)} - {z2}^{(1)}} \right)}}}}\end{matrix} \right.$

[0171] From this we immediately deduce: $\begin{matrix}\left\{ \begin{matrix}{x = {\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)y}} \\{\quad {{z1} = {\left( {{\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)y}}} \\{\quad {{z2} = {\left( {{\frac{r2}{k2}s} + 1} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)y}}}\end{matrix} \right. & (6)\end{matrix}$

[0172] i.e, putting $\frac{mi}{ki} = {Ti}^{2}$

[0173] and ${\frac{ri}{ki} = {2{DiTi}}},{i = {1.2:}}$

$\quad\left\{ \begin{matrix}{\quad {x = {y + {2\left( {{D1T1} + {D2T2}} \right)y^{(1)}} + {\left( {{Ti}^{2} + {T2}^{2} + {4{D1D2T2}}} \right)y^{(2)}} +}}} \\{\quad {{2\left( {{D1T1T2}^{2} + {D2T2T1}^{2}} \right)y^{(3)}} + {\left( {{T1}^{2}{T2}^{2}} \right)y^{(4)}}}} \\{{z1} = {y + {2\left( {{D1T1} + {D2T2}} \right)y^{(1)}} + {\left( {{T2}^{2} + {4{D1D2T1T2}}} \right)y^{(2)}} + {\left( {2{D1T1T2}^{2}} \right)y^{(3)}}}} \\{{z2} = {y + {2\left( {{D1T1} + {D2T2}} \right)y^{(1)}} + {\left( {{T1}^{2} + {4{D1D2T1T2}}} \right)y^{(2)}} + {\left( {2{D2T2T1}^{2}} \right){y^{(3)}.}}}}\end{matrix} \right.$

[0174] The expression for y, or more precisely the expressions for y,y⁽¹⁾, y⁽²⁾, y⁽³⁾, y⁽⁴⁾ and y⁽⁵⁾ are deduced therefrom by inverting thesystem obtained on the basis of x, z1, z2, x⁽¹⁾, z1 ⁽¹⁾, z2 ⁽¹⁾.

[0175] We deduce therefrom that, to perform a displacement from x0 atthe instant t0 to x1 at the instant t1, with the auxiliary masses atrest at t0 and t1, it is sufficient to construct a reference trajectoryfor y with the initial and final conditions

[0176] y(t0)=x0, y(t1)=x1 and all the derivativesy^((k))(t0)=y^((k))(t1)=0, k varying from 1 to 6 or more if necessary,and to deduce therefrom the reference trajectories of the main andauxiliary masses, as well as of the force F to be applied to the motor.

[0177] In this case, the force F satisfies the relation:${F(t)} = {\left\lbrack {{\left( {{Ms}^{2} + {\left( {{r1} + {r2}} \right)s} + \left( {{k1} + {k2}} \right)} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)} - {\left( {{r1s} + {k1}} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)} - {\left( {{r2s} + {k2}} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)}} \right\rbrack {y.}}$

[0178] Furthermore, the model for three auxiliary masses MAi (p=3) [seeFIG. 1], may be written, as earlier: $\left\{ \begin{matrix}\begin{matrix}\begin{matrix}{\quad {{Mx}^{(2)} = \quad {F - {{k1}\left( {x - {z1}} \right)} - {{r1}\left( {x^{(1)} - {z1}^{(1)}} \right)}}}} \\{\quad {{- {{k2}\left( {x - {z2}} \right)}} - {{r2}\left( {x^{(1)} - {z2}^{(1)}} \right)} - {{k3}\left( {x - {z3}} \right)} - {{r3}\left( {x^{(1)} - {z3}^{(1)}} \right)}}}\end{matrix} \\{\quad {{m1z1}^{(2)} = \quad {{{k1}\left( {x - {z1}} \right)} + {{r1}\left( {x^{(1)} - {z1}^{(1)}} \right)}}}}\end{matrix} \\{\quad {{m2z2}^{(2)} = \quad {{{k2}\left( {x - {z2}} \right)} + {{r2}\left( {x^{(1)} - {z2}^{(1)}} \right)}}}} \\{\quad {{m3z3}^{(2)} = \quad {{{k3}\left( {x - {z3}} \right)} + {{r3}\left( {x^{(1)} - {z3}^{(1)}} \right)}}}}\end{matrix} \right.$

[0179] From this we immediately deduce: $\begin{matrix}\left\{ \begin{matrix}{\quad {x = {\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)\left( {{\frac{m3}{k3}s^{2}} + {\frac{r3}{k3}s} + 1} \right)y}}} \\{\quad {{z1} = {\left( {{\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)\left( {{\frac{m3}{k3}s^{2}} + {\frac{r3}{k3}s} + 1} \right)y}}} \\{\quad {{z2} = {\left( {{\frac{r2}{k2}s} + 1} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m3}{k3}s^{2}} + {\frac{r3}{k3}s} + 1} \right)y}}} \\{\quad {{z3} = {\left( {{\frac{r3}{k3}s} + 1} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)y}}}\end{matrix} \right. & (7)\end{matrix}$

[0180] We proceed as earlier in order to determine the values as afunction of time of the different variables and in particular of theforce F, the latter satisfying the expression: $\begin{matrix}{{F(t)} = \quad \left\lbrack {{\left( {{Ms}^{2} + {\left( {{r1} + {r2} + {r3}} \right)s} + \left( {{k1} + {k2} + {k3}} \right)} \right).} \cdot} \right.} \\{\quad {{\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)\left( {{\frac{m3}{k3}s^{2}} + {\frac{r3}{k3}s} + 1} \right)} -}} \\{\quad {{\left( {{r1s} + {k1}} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right)\left( {{\frac{m3}{k3}s^{2}} + {\frac{r3}{k3}s} + 1} \right)} -}} \\{\quad {{\left( {{r2s} + {k2}} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m3}{k3}s^{2}} + {\frac{r3}{k3}s} + 1} \right)} -}} \\{\quad {\left. \left( {{r3s} + {k3}} \right)\left( {{\frac{m1}{k1}s^{2}} + {\frac{r1}{k1}s} + 1} \right)\left( {{\frac{m2}{k2}s^{2}} + {\frac{r2}{k2}s} + 1} \right) \right\rbrack{y.}}}\end{matrix}$

[0181] Represented in FIGS. 8 to 13 are the values respectively of thevariables y, x, z1, z2, z3 and F as a function of time t for aparticular example of the embodiment of FIG. 1, z1 to z3 being thedisplacements of the auxiliary masses MA1, MA2 and MA3 respectively. Thevariables y, x, z1, z2 and z3 are expressed in meters (m) and the forceF in Newtons (N).

[0182] This example is such that:

[0183] M=5 kg;

[0184] m1=0.1 kg;

[0185] m2=0.01 kg;

[0186] m3=0.5 kg;

[0187] k1=m1(5.2π)², k2=m2(4.2π)², k3=m3(6.2π)², corresponding tonatural frequencies of 5, 4 and 6 Hz respectively;

[0188] r1=0.3{square root}{square root over (k1m1)}, r2=0.2{squareroot}{square root over (k2m2)}, r3=0.15{square root}{square root over(k3m3)}; corresponding to normalized dampings of 0.3, 0.2 and 0.15respectively;

[0189] t1−t0=0.34 s; and

[0190] x1−x0=40 mm.

[0191] Additionally, in a third embodiment represented in FIG. 2, thesystem S2 comprises the moveable unit 4, the base 2 which is mountedelastically with respect to the floor S and an auxiliary mass MA whichis linked by way of an elastic link eA of standard type to said base 2.

[0192] In this case, the variables of the system are the positions x, xBand zA of the moveable unit 4, of the base B and of the auxiliary massMA, which satisfy the relations: $\begin{matrix}\left\{ \begin{matrix}{\quad {x = {\left\lbrack {{\left( {{mAs}^{2} + {rAs} + {kA}} \right) \cdot \left( {{mBs}^{2} + {\left( {{rA} + {rB}} \right)s} + \left( {{kA} + {kB}} \right)} \right)} - \left( {{rAs} + {kA}} \right)^{2}} \right\rbrack \cdot y}}} \\{\quad {{xB} = {- {My}^{(2)}}}} \\{\quad {{zA} = {- {M\left( {{rAy}^{(3)} + {kAy}^{(2)}} \right)}}}}\end{matrix} \right. & (8)\end{matrix}$

[0193] in which:

[0194] M, mB and mA are the masses respectively of the moveable unit 4,of the base 2 and of the auxiliary mass MA;

[0195] rA and rB are the dampings respectively of the auxiliary mass MAand of the base 2;

[0196] kA and kB are the stiffnesses respectively of the auxiliary massMA and of the base 2; and

[0197] s=d/dt.

[0198] Specifically, the dynamic model of the system S2 may be written:$\begin{matrix}{\quad\left\{ \begin{matrix}{\quad {{Mx}^{(2)} = F}} \\{{mBxB}^{(2)} = {{- F} - {kBxB} - {rBxB}^{(1)} - {k\left( {{xB} - {zA}} \right)} - {{rA}\left( {{xB}^{(1)} - {zA}^{(1)}} \right)}}} \\{\quad {{mzA}^{(2)} = {{{kA}\left( {{xB} - {zA}} \right)} + {{rA}\left( {{xB}^{(1)} - {zA}^{(1)}} \right)}}}}\end{matrix} \right.} & (9)\end{matrix}$

[0199] The intermediate variable must satisfy: x=P(s)y, xB=PB(s)y andzA=Pz(s)y with $s = {\frac{}{t}.}$

[0200] Substituting these expressions into (9), we obtain:$\quad\left\{ \begin{matrix}{\quad {F = {{Ms}^{2}{P(s)}y}}} \\{{\left( {{mBs}^{2} + {\left( {{rA} + {rB}} \right)s} + \left( {{kA} + {kB}} \right)} \right){{PB}(s)}} = {{{- {Ms}^{2}}{P(s)}} + {\left( {{rAs} + {kA}} \right){{Pz}(s)}}}} \\{\quad {{\left( {{mAs}^{2} + {rAs} + {kA}} \right){{Pz}(s)}} = {\left( {{rAs} + {kA}} \right){{{PB}(s)}.}}}}\end{matrix} \right.$

[0201] On eliminating Pz from the last equation, it follows that:

[(mAs ² +rAs+kA)(mBs ²+(rA+rB)s+(kA+kB))−(rAs+kA)² ]PB=−(mAs ²+rAs+kA)Ms ² P

[0202] from which we derive: $\quad\left\{ \begin{matrix}{P = {{\left( {{mAs}^{2} + {rAs} + {kA}} \right)\left( {{mBs}^{2} + {\left( {{rA} + {rB}} \right)s} + \left( {{kA} + {kB}} \right)} \right)} - \left( {{rAs} + {kA}} \right)^{2}}} \\{\quad {{PB} = {- {Ms}^{2}}}} \\{\quad {{Pz} = {- {{Ms}^{2}\left( {{rAs} + {kA}} \right)}}}}\end{matrix} \right.$

[0203] thus making it possible to obtain the aforesaid expressions (8).

[0204] The values as a function of time of the different variables, andin particular the force F, are then obtained as before.

[0205] In this case, said force F satisfies the expression:

F(t)=M[(mAs ² +rAs+kA)(mBs ²+(rA+rB)s+(kA+kB))−(rAs+kA)² ]y ⁽²⁾.

[0206] In a fourth and last embodiment (not represented), the system isformed of the moveable unit 4, of the base 2 and of an auxiliary mass MCwhich is tied elastically to said moveable unit 4.

[0207] In this case, the variables of the system are the positions x, xBand zC respectively of the moveable unit 4, of the base 2 and of theauxiliary mass MC, which satisfy the relations:$\quad\left\{ \begin{matrix}{\quad {x = \left\lbrack {\left( {{mCs}^{2} + {rCs} + {kC}} \right) \cdot \left. \left( {{mBs}^{2} + {rBs} + {kB}} \right. \right\rbrack \cdot y} \right.}} \\{{xB} = {\left\lbrack {{\left( {{mCs}^{2} + {rCs} + {kC}} \right) \cdot \left( {{Ms}^{2} + {rCs} + {kC}} \right)} - \left( {{rCs} + {kC}} \right)^{2}} \right\rbrack \cdot y}} \\{\quad {{zC} = {\left( {{rCs} + {kC}} \right){\left( {{mBs}^{2} + {rBs} + {kB}} \right) \cdot y}}}}\end{matrix} \right.$

[0208] in which:

[0209] M, mB and mC are the masses respectively of the moveable unit 4,of the base 2 and of the auxiliary mass MC;

[0210] rB and rC are the dampings respectively of the base 2 and of theauxiliary mass MC;

[0211] kB and kC are the stiffnesses respectively of the base 2 and ofthe auxiliary mass MC; and

[0212] s=d/dt.

[0213] Specifically, the dynamic model of this system may be written:$\begin{matrix}\left\{ \begin{matrix}{{Mx}^{(2)} = {F - {{kC}\left( {x - {zC}} \right)} - {{rC}\left( {x^{(1)} - {zC}^{(1)}} \right)}}} \\{\quad {{mBxB}^{(2)} = {{- F} - {kBxB} - {rBxB}^{(1)}}}} \\{\quad {{mCzC}^{(2)} = {{{kC}\left( {x - {zC}} \right)} + {{rC}\left( {x^{(1)} - {zC}^{(1)}} \right)}}}}\end{matrix} \right. & (10)\end{matrix}$

[0214] By using, as in the foregoing, the polynomial representation ofthe variable ${s = \frac{}{t}},$

[0215] the system (10) becomes: $\left\{ {\begin{matrix}{{\left( {{Ms}^{2} + {rCs} + {kC}} \right)x} = {F + {\left( {{rCs} + {kC}} \right){zC}}}} \\{\quad {{\left( {{mBs}^{2} + {rBs} + {kB}} \right){xB}} = {- F}}} \\{\quad {{\left( {{mCs}^{2} + {rCs} + {kC}} \right){zC}} = {\left( {{rCs} + {kC}} \right)x}}}\end{matrix},} \right.$

[0216] which, together with the expressions for each of the variables asa function of the intermediate variable (and of its derivatives)x=P(s)y, xB=PB(s)y, zC=Pz(s)y, finally gives:$\quad\left\{ \begin{matrix}{\quad {P = {\left( {{mCs}^{2} + {rCs} + {kC}} \right)\left( {{mBs}^{2} + {rBs} + {kB}} \right)}}} \\{\quad {{PB} = {{\left( {{mCs}^{2} + {rCs} + {kC}} \right)\left( {{Ms}^{2} + {rCs} + {kC}} \right)} - \left( {{rCs} + {kC}} \right)^{2}}}} \\{\quad {{Pz} = {\left( {{rCs} + {kC}} \right)\left( {{mBs}^{2} + {rBs} + {kB}} \right)}}}\end{matrix} \right.$

[0217] The construction of the reference trajectories of y, and then ofx, xB, zC and F is done as indicated earlier.

[0218] In this case, the force F satisfies:

F(t)=−(mBs ² +rBs+kB)[(mCs ² +rCs+kC)(Ms ² +rCs+kC)−(rCs+kC)² ]y.

[0219] A method in accordance with the invention will now be describedwhich makes it possible to determine in a general and fast manner theexpressions defined in the aforesaid step b) of the process inaccordance with the invention, for linear systems of the form:$\begin{matrix}{{\sum\limits_{j = 1}^{p}\quad {Ai}},{{{j(s)}{xj}} = {biF}},{i = 1},\ldots \quad,p} & (11)\end{matrix}$

[0220] where the Ai,j(s) are polynomials of the variable s, which, inthe case of coupled mechanical systems, are of degree less than or equalto 2 and where one at least of the coefficients bi is non-zero. F is thecontrol input which, in the above examples, is the force produced by theactuator 5.

[0221] Accordingly, according to the invention, in step b), thefollowing operations are carried out:

[0222] the different variables xi of said system (for example S1 or S2),i going from 1 to p, p being an integer greater than or equal to 2, eachbeing required to satisfy a first expression of the form:${{{xi} = {\sum\limits_{j = 0}^{j = r}{pi}}},{j \cdot y^{(j)}},}\quad$

[0223] the y^((j)) being the derivatives of order j of the intermediatevariable y, r being a predetermined integer and the pi, j beingparameters to be determined, a second expression is obtained by puttingy^((j))=s^((j)).y:${xi} = {{\left( {{\sum\limits_{j = 0}^{j = r}{pi}},{j \cdot s^{j}}} \right)\quad y} = {{{Pi}(s)} \cdot y}}$

[0224] a third expression of vectorial type is defined on the basis ofthe second expressions relating to the different variables xi of thesystem: X = P ⋅ y $\left\{ \begin{matrix}{P = \begin{pmatrix}{P1} \\\vdots \\P_{P}\end{pmatrix}} \\{X = \begin{pmatrix}{x1} \\\vdots \\{xp}\end{pmatrix}}\end{matrix} \right.$

[0225] comprising the vectors

[0226] said vector P is calculated, replacing X by the value P.y in thefollowing expressions: $\quad\left\{ \begin{matrix}{{B^{T} \cdot {A(s)} \cdot {P(s)}} = {{Op} - 1}} \\{{{bp} \cdot F} = \left( {{\sum\limits_{j = 1}^{j = p}{Ap}},{{j(s)} \cdot {{Pj}(s)} \cdot Y}} \right)}\end{matrix} \right.$

[0227] in which:

[0228] B^(T) is the transpose of a matrix B of size px(p 1) and of rankp−1, such that B^(T)b=Op−1;

[0229] bp is the p-th component of the vector b; and

[0230] Op−1 is a zero vector of dimension (p−1);

[0231] the values of the different parameters pi,j are deduced from thevalue thus calculated of the vector P; and

[0232] from these latter values are deduced the values of the variablesxi as a function of the intermediate variable y and of its derivatives,on each occasion using the corresponding first expression.

[0233] The aforesaid method is now justified.

[0234] Let us denote by A(s) the matrix of size pxp whose coefficientsare the polynomials Ai, j(s), i, j=1, . . . , p, i.e.:${{{A(s)} = \left( {\begin{matrix}{{A1},{1(s)}} \\\vdots \\{{Ap},{1(s)}}\end{matrix}\begin{matrix}\cdots \\\quad \\\cdots\end{matrix}\begin{matrix}{{A1},{p(s)}} \\\vdots \\{{Ap},{p(s)}}\end{matrix}} \right)},{X = {{\begin{pmatrix}{x1} \\\vdots \\{xp}\end{pmatrix}\quad {and}\quad b} = \begin{pmatrix}{b1} \\\vdots \\{bp}\end{pmatrix}}}}\quad$

[0235] Without loss of generality, it can be assumed that the rank ofA(s) is equal to p (otherwise, the system is written together with itsredundant equations and it is sufficient to eliminate the dependentequations) and that bp≠0. There then exists a matrix B of size px(p−1)and of rank p−1such that:

B ^(T) b= 0 p−1

[0236] where T represents transposition and 0p−1 the vector of dimensionp−1, all of whose components are zero. The system (11) premultiplied byB^(T) then becomes: $\begin{matrix}{{{B^{T}{A(s)}X} = {{Op} - 1}},{{bpF} = {\sum\limits_{j = 1}^{p}\quad {Ap}}},{j \times {j.}}} & (12)\end{matrix}$

[0237] As indicated earlier, an intermediate variable y is characterizedin that all the components of the vector X can be expressed as afunction of y and of a finite number of its derivatives. For acontrollable linear system, such an output always exists and thecomponents of X can be found in the form of linear combinations of y andof its derivatives, i.e.:${{xi} = {\sum\limits_{j = 0}^{r}\quad {pi}}},{jy}^{(j)}$

[0238] where y^((j)) is the derivative of order j of y with respect totime and where the pi,j are real numbers which are not all zero, oralternatively:${{xi} = {{\left( {{\sum\limits_{j = 0}^{r}\quad {pi}},{js}^{j}} \right)y} = {{{Pi}(s)}y}}},{i = 1},\ldots \quad,{p.}$

[0239] We shall calculate the vector ${{P(s)} = \begin{pmatrix}{{P1}(s)} \\\vdots \\{{Pp}(s)}\end{pmatrix}},$

[0240] by replacing X by its value P(s)y in (12): $\begin{matrix}{{{bpF} = {\sum\limits_{j = 1}^{p}\quad {Ap}}},{{j(s)}{{Pj}(s)}{y.}}} & (13)\end{matrix}$

[0241] Consequently, P belongs to the kernel of the matrix B^(T)A(s) ofdimension 1, since B is of rank p−1 and A(s) of rank p. To calculate P,let us denote by A1(s), . . . , Ap(s) the columns of the matrix A(s) andÂ(s) the matrix of size (p−1)×(p−1) defined by:

Â(s)=(A 2(s), . . . , Ap(s)).

[0242] Let us also denote by P(s) the vector of dimension p−1 definedby: ${\hat{P}(s)} = {\begin{pmatrix}{{P2}(s)} \\\vdots \\{{Pp}(s)}\end{pmatrix}.}$

[0243] Let us rewrite (13) in the formB^(T)A1(s)P1(s)+B^(T)Â(s){circumflex over (P)}(s)=0p−1 or alternativelyB^(T)Â(s){circumflex over (P)}(s)=−B^(T)A1(s)P1(s). Since the matrixB^(T)Â(s) is invertible, we have:

{circumflex over (P)}(s)=−(B ^(T) Â(s))⁻¹ B ^(T) A 1(s)P 1(s)

[0244] i.e.: $\begin{matrix}{{\hat{P}(s)} = {{- \frac{1}{\det \left( {B^{T}{\hat{A}(s)}} \right)}}\left( {{co}\left( {B^{T}{\hat{A}(s)}} \right)} \right)^{T}B^{T}{{A1}(s)}{{P1}(s)}}} & (14)\end{matrix}$

[0245] where co(B^(T)Â(s)) is the matrix of the cofactors of B^(T)Â(s).

[0246] From this we immediately deduce that it is sufficient to choose:$\begin{matrix}\left\{ \begin{matrix}{\quad {{{P1}(s)} = {\det \left( {B^{T}{\hat{A}(s)}} \right)}}} \\{{\hat{P}(s)} = {{- \left( {{co}\left( {B^{T}{\hat{A}(s)}} \right)} \right)^{T}}B^{T}{{A1}(s)}}}\end{matrix} \right. & (15)\end{matrix}$

[0247] this completing the calculation of the vector P(s).

[0248] It will be observed that if the Ai,j(s) are polynomials of degreeless than or equal to m, the degree of each of the components of P isless than or equal to mp. Specifically, in this case, the degree of thedeterminant det (B^(T)Â(s)) is less than or equal to (p−1)m and thedegree of each of the rows of (co (B^(T)Â(s)))^(T)B^(T) A1(s), using thefact that the degree of a product of polynomials is less than or equalto the sum of the degrees, is less than or equal to (p−1)m+m=pm, hencethe aforesaid result.

[0249] In all the examples presented earlier, which model mechanicalsubsystems, we have m=2.

[0250] It may easily be verified that this general method yields thesame calculations for P as in each of the examples already presentedhereinabove.

[0251] We shall return to certain of the examples dealt with earlier andshow how the calculation of the variable y makes it possible to achievepassive isolation of the elastic modes.

[0252] In all these examples, the trajectories are generated on thebasis of polynomial trajectories of the intermediate value y, which areobtained through interpolation of the initial and final conditions.Furthermore, we are interested only in the particular case where thesystem is at rest at the initial and final instants, thereby making itpossible to establish simple and standard formulae which depend only onthe degree of the polynomial.

[0253] In the simplest case, where the initial and final derivatives ofy are zero up to order 4, the sought-after polynomial is of degree 9:$\quad\left\{ \begin{matrix}{{y({t0})} = {y0}} & {{y^{(1)}({t0})} = 0} & {{y^{(2)}({t0})} = 0} & {{y^{(3)}({t0})} = 0} & {{y^{(4)}({t0})} = 0} \\{{y({t1})} = {y1}} & {{y^{(1)}({t1})} = 0} & {{y^{(2)}({t1})} = 0} & {{y^{(3)}({t1})} = 0} & {{y^{(4)}({t1})} = 0}\end{matrix} \right.$

[0254] which gives:

y(t)=y 0+(y 1−y 0)σ⁵(126−420σ+540σ²−315σ³+70σ⁴), $\begin{matrix}{\sigma = \left( \frac{t - {t0}}{{t1} - {t0}} \right)} & (16)\end{matrix}$

[0255] If we ask for a polynomial such that the initial and finalderivatives are zero up to order 5, the sought-after polynomial is ofdegree 11:

y(t)=y 0+(y 1−y 0)σ⁶(462−1980σ+3465σ²−3080σ³+1386σ⁴−252σ⁵)

[0256] still with σ defined as in (16).

[0257] If we ask for a polynomial such that the initial and finalderivatives are zero up to order 6, the sought-after polynomial is ofdegree 13:

y(t)=y 0+(y 1−y 0)σ⁷(1716−9009σ+20020σ²−24024σ³+16380σ⁴−6006σ⁵+924σ⁶).

1. A process for displacing a moveable unit (4) on a base (2), saidmoveable unit (4) being displaced linearly according to a predetermineddisplacement under the action of a controllable force (F), wherein: a)equations are defined which: illustrate a dynamic model of a systemformed by elements (2, 4, MA, MA1, MA2, MA3), of which said moveableunit (4) is one, which are brought into motion upon a displacement ofsaid moveable unit (4); and comprise at least two variables, of whichthe position of said moveable unit (4) is one; b) all the variables ofthis system, together with said force (F), are expressed as a functionof one and the same intermediate variable y and of a specified number ofderivatives as a function of time of this intermediate variable, saidforce (F) being such that, applied to said moveable unit (4), itdisplaces the latter according to said specified displacement andrenders all the elements of said system immobile at the end of saiddisplacement; c) the initial and final conditions of all said variablesare determined; d) the value as a function of time of said intermediatevariable is determined from the expressions for the variables defined instep b) and said initial and final conditions; e) the value as afunction of time of said force is calculated from the expression for theforce, defined in step b) and said value of the intermediate variable,determined in step d); and f) the value thus calculated of said force(F) is applied to said moveable unit (4).
 2. The process as claimed inclaim 1, wherein, in step a), the following operations are carried out:the variables of the system are denoted xi, i going from 1 to p, p beingan integer greater than or equal to 2, and the balance of the forces andof the moments is expressed, approximating to first order if necessary,in the so-called polynomial matrix form: A(s)X=bF with: A(s) matrix ofsize p×p whose elements Aij(s) are polynomials of the variable s=d/dt;${X\quad {the}\quad {{vector}\begin{pmatrix}{x1} \\\vdots \\{xp}\end{pmatrix}}};$

b the vector of dimension p; and F the force exerted by a means ofdisplacing the moveable unit and in that, in step b), the followingoperations are carried out: the different variables xi of said system, igoing from 1 to p, each being required to satisfy a first expression ofthe form:${{xi} = {\sum\limits_{j = 0}^{j = r}{pi}}},{j \cdot y^{(j)}},$

the y^((j)) being the derivatives of order j of the intermediatevariable y, r being a predetermined integer and the pi,j beingparameters to be determined, a second expression is obtained by puttingy^((j))=s^(j).y:${{xi} = {{\left( {{\sum\limits_{j = 0}^{j = r}{pi}},{j \cdot s^{j}}} \right)y} = {{{Pi}(s)} \cdot y}}},$

a third expression of vectorial type is defined on the basis of thesecond expressions relating to the different variables xi of the system(S1, S2): X=P.y comprising the vector $P = \begin{pmatrix}{P1} \\\vdots \\{Pp}\end{pmatrix}$

said vector P is calculated, by replacing X by the value P.y in thefollowing system: $\quad\left\{ \begin{matrix}{{B^{T} \cdot {A(s)} \cdot {P(s)}} = {{Op} - 1}} \\{{{{bp} \cdot F} = {\sum\limits_{j = 1}^{j = p}{Ap}}},{{j(s)} \cdot {{Pj}(s)} \cdot y}}\end{matrix} \right.$

in which: B^(T) is the transpose of a matrix B of size px(p−1) such thatB^(T)b=Op−1; bp is the p-th component of the vector b previouslydefined; and Op−1 is a zero vector of dimension (p−1); the values of thedifferent parameters pi,j are deduced from the value thus calculated ofthe vector P; and from these latter values are deduced the values of thevariables xi as a function of the intermediate variable y and of itsderivatives, on each occasion using the corresponding first expression.3. The process as claimed in claim 1, wherein, in step d), a polynomialexpression for the intermediate variable y is used to determine thevalue of the latter.
 4. The process as claimed in claim 3, wherein, theinitial and final conditions of the different variables of the system,together with the expressions defined in step b), are used to determinethe parameters of the polynomial expression for the intermediatevariable y.
 5. The process as claimed in claim 1 for displacing amoveable unit (4) on a base (2) which is mounted elastically withrespect to the floor (S) and which may be subjected to linear andangular motions, wherein the variables of the system are the linearposition x of the moveable unit, the linear position xB of the base andthe angular position θz of the base, which satisfy the relations:$\quad\left\{ \begin{matrix}{x = \quad {y + {\left( {\frac{rB}{kB} + \frac{r\quad \theta}{k\quad \theta}} \right)y^{(1)}} + {\left( {\frac{mB}{kB} + \frac{{rBr}\quad \theta}{{kBk}\quad \theta} + \frac{J}{k\quad \theta}} \right)y^{(2)}} +}} \\{\quad {{\left( {\frac{rBJ}{{kBk}\quad \theta} + \frac{{mBr}\quad \theta}{{kBk}\quad \theta}} \right)y^{(3)}} + {\frac{mBJ}{{kBk}\quad \theta}y^{(4)}}}} \\{{xB} = {{- \frac{m}{kB}}\left( {{\frac{J}{k\quad \theta}y^{(4)}} + {\frac{r\quad \theta}{k\quad \theta}y^{(3)}} + y^{(2)}} \right)}} \\{{\theta \quad z}\quad = {{- d}\quad \frac{m}{k\quad \theta}\left( {{\frac{mB}{kB}y^{(4)}} + {\frac{rB}{kB}y^{(3)}} + y^{(2)}} \right)}}\end{matrix} \right.$

in which: m is the mass of the moveable unit; mB, kB, kθ, rB, rθ arerespectively the mass, the linear stiffness, the torsional stiffness,the linear damping and the torsional damping of the base; J is theinertia of the base with respect to a vertical axis; d is the distancebetween the axis of translation of the center of mass of the moveableunit and that of the base; and y⁽¹⁾, y⁽²⁾, y⁽³⁾ and y⁽⁴⁾ arerespectively the first to fourth derivatives of the variable y.
 6. Theprocess as claimed in claim 1 for displacing on a base a moveable unit(4) on which are elastically mounted a number p of auxiliary masses MAi,p being greater than or equal to 1, i going from 1 to p, wherein thevariables of the system are the position x of the moveable unit (4) andthe positions zi of the p auxiliary masses MAi, which satisfy therelations: $\quad\left\{ \begin{matrix}{x = {\left( {\prod\limits_{i = 1}^{p}\left( {{\frac{mi}{ki}s^{2}} + {\frac{ri}{ki}s} + 1} \right)} \right) \cdot y}} \\{{zi} = {\left( {\prod\limits_{\underset{j \neq i}{j = 1}}^{p}\left( {{\frac{mj}{kj}s^{2}} + {\frac{rj}{kj}s} + 1} \right)} \right) \cdot \left( {{\frac{ri}{ki}s} + 1} \right) \cdot y}}\end{matrix} \right.$

in which: Π illustrates the product of the associated expressions; mi,zi, ki and ri are respectively the mass, the position, the stiffness andthe damping of an auxiliary mass MAi; mj, kj and rj are respectively themass, the stiffness and the damping of an auxiliary mass MAj; ands=d/dt.
 7. The process as claimed in claim 1 for displacing a moveableunit (4) on a base (2) which is mounted elastically with respect to thefloor (S) and on which is elastically mounted an auxiliary mass (MA),wherein the variables of the system are the positions x, xB and zArespectively of the moveable unit (4), of the base (2) and of theauxiliary mass (MA), which satisfy the relations:$\quad\left\{ \begin{matrix}{x = \quad \left\lbrack {{\left( {{mAs}^{2} + {rAs} + {kA}} \right) \cdot \left( {{mBs}^{2} + {\left( {{rA} + {rB}} \right)s} + \left( {{kA} + {kB}} \right)} \right)} -} \right.} \\{\left. \quad \left( {{rAs} + {kA}} \right)^{2} \right\rbrack \cdot y} \\{{xB} = {- {My}^{(2)}}} \\{{zA} = {- {M\left( {{rAy}^{(3)} + {kAy}^{(2)}} \right)}}}\end{matrix} \right.$

in which: M, mB and mA are the masses respectively of the moveable unit(4), of the base (2) and of the auxiliary mass (MA); rA and rB are thedampings respectively of the auxiliary mass (MA) and of the base (2); kAand kB are the stiffnesses respectively of the auxiliary mass (MA) andof the base (2); and s=d/dt.
 8. The process as claimed in claim 1 fordisplacing on a base mounted elastically with respect to the floor, amoveable unit on which is elastically mounted an auxiliary mass, whereinthe variables of the system are the positions x, xB and zC respectivelyof the moveable unit, of the base and of the auxiliary mass, whichsatisfy the relations: $\quad\left\{ \begin{matrix}{x = \left\lbrack {\left( {{mCs}^{2} + {rCs}\quad + {kC}} \right) \cdot \left( {{mBs}^{2} + {rBs} + {kB}} \right\rbrack \cdot y} \right.} \\{{xB} = {\left\lbrack {{\left( {{mCs}^{2} + {rCs} + {kC}} \right) \cdot \left( {{Ms}^{2} + {rCs}\quad + {kC}} \right)} - \left( {{rCs} + {kC}} \right)^{2}} \right\rbrack \cdot y}} \\{{zC} = {\left( {{rCs}\quad + {kC}} \right) \cdot \left( {{mBs}^{2} + {rBs} + {kB}} \right) \cdot y}}\end{matrix} \right.$

in which: M, mB and mC are the masses respectively of the moveable unit,of the base and of the auxiliary mass; rB and rC are the dampingsrespectively of the base and of the auxiliary mass; kB and kC are thestiffnesses respectively of the base and of the auxiliary mass; ands=d/dt.
 9. A device comprising: a base (2); a moveable unit (4) whichmay be displaced linearly on said base (2); and a controllable actuator(5) able to apply a force (F) to said moveable unit (4) with a view toits displacement on said base (2), wherein it furthermore comprisesmeans (6) which implement steps a) to e) of the process specified underclaim 1, so as to calculate a force (F) which may be applied to saidmoveable unit (4), and which determine a control command and transmit itto said actuator (5) so that it applies the force (F) thus calculated tosaid moveable unit (4).